Find Range of \(f(x)=\cos([x])\)

📝 Question

Let:

\[ f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\to \mathbb{R}, \quad f(x)=\cos([x]) \]

where \([x]\) denotes the greatest integer function. Find the range of \(f\).

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✅ Solution

🔹 Step 1: Find possible values of \([x]\)

\[ -\frac{\pi}{2} \approx -1.57,\quad \frac{\pi}{2} \approx 1.57 \]

So \(x \in (-1.57, 1.57)\)

Thus possible integer values of \([x]\) are:

\[ [x] = -2, -1, 0, 1 \] —

🔹 Step 2: Compute function values

\[ f(x)=\cos([x]) \]

So values are:

\[ \cos(-2),\quad \cos(-1),\quad \cos(0),\quad \cos(1) \] —

🔹 Step 3: Simplify using identity

\[ \cos(-\theta)=\cos(\theta) \]

Thus,

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So distinct values are:

\[ \cos 2,\quad \cos 1,\quad 1 \] —

🎯 Final Answer

\[ \boxed{\{\cos 2,\ \cos 1,\ 1\}} \]

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🚀 Exam Shortcut

• Convert interval into decimals
• Find all possible integer values of \([x]\)
• Evaluate function at those integers
• Use \(\cos(-x)=\cos x\)